Why Open-Source?

A Little Story..

Once upon a time, there was a young, naive and yet brilliant, scientist, who had a thought..

..the young scientist then came across an intriguing paper about the discrepancy in the PK of an antifungal drug, called voriconazole, when administered to adults versus children https://link.springer.com/article/10.1007%2Fs40262-014-0181-y.

What Open-Source Offers:

  • Free accessibility
  • Easy communication
  • Transparency
  • Reproducibility

Example of open-source PBPK modeling and simulation software is PK-Sim/MoBi http://www.open-systems-pharmacology.org/

Why R/mrgsolve?

  • Fun!

General PBPK Workflow

Let’s Start Coding..

1. Playing with R and tidyverse

Open a new Rscript file and play a little with R and the package tidyverse..

2. PBPK Modeling and Simulation

a. mrgsolve

Elmokadem, Ahmed, Matthew M. Riggs, and Kyle T. Baron. 2019. “Quantitative Systems Pharmacology and Physiologically-Based Pharmacokinetic Modeling With Mrgsolve: A Hands-On Tutorial.” CPT: Pharmacometrics & Systems Pharmacology, October. https://doi.org/10.1002/psp4.12467.

# load libraries
library(tidyverse)
library(mrgsolve)

b. Simple PBPK Model

Model structure

Equations

  • Non-eliminating compartment (Muscle)

\[\frac{dA_{mu}}{dt}=Q_{mu}(C_{art}-\frac{C_{mu}}{\frac{Kp_{mu}}{BP}})\]

  • Eliminating compartment (Liver):

\[\frac{dA_{li}}{dt}=Q_{li}(C_{art}-\frac{C_{li}}{\frac{Kp_{li}}{BP}})-Cl_{li}.f_{u}.\frac{C_{li}}{\frac{Kp_{li}}{BP}}\]

  • Arterial blood compartment:

\[\frac{dA_{art}}{dt}=Q_{lu}(\frac{C_{lu}}{\frac{Kp_{lu}}{BP}}-C_{art})\]

  • Venous blood compartment:

\[\frac{dA_{ven}}{dt}=\sum_{T\neq lu} (Q_T.\frac{C_T}{\frac{Kp_T}{BP}}) − Q_{lu}.C_{ven}\]

  • Lungs:

\[\frac{dA_{lu}}{dt}=Q_{lu}(C_{ven} − \frac{C_{lu}}{\frac{Kp_{lu}}{BP}})\]

Tasks

  1. Use ../model/simplePBPK.cpp file to build a simple PBPK model.

  2. Use script.R script to compile the model and run a simple simulation (chunks 1 and 2).

b. Voriconazole PBPK Model

References

Zane NR, Thakker DR. A physiologically based pharmacokinetic model for voriconazole disposition predicts intestinal first-pass metabolism in children. Clin Pharmacokinet. 2014;53: 1171–1182
https://link.springer.com/article/10.1007%2Fs40262-014-0181-y

ICRP Publication 89 http://www.icrp.org/publication.asp?id=ICRP%20Publication%2089

Utsey, Kiersten, Madeleine S. Gastonguay, Sean Russell, Reed Freling, Matthew M. Riggs, and Ahmed Elmokadem. 2020. “Quantification of the Impact of Partition Coefficient Prediction Methods on PBPK Model Output Using a Standardized Tissue Composition.” Drug Metabolism and Disposition: The Biological Fate of Chemicals, July. https://doi.org/10.1124/dmd.120.090498.

Model structure

Tasks

Note: for the following tasks use script.R

  1. Explore and compile typical adult male PBPK model ../model/voriPBPK.cpp
modA <- mread("../model/voriPBPK")
  1. Use calcKp_PT.R function to calculate voriconzole tissue:plasma partition coefficients according to Poulin and Theil method https://www.ncbi.nlm.nih.gov/pubmed/11782904.
source("calcKp_PT.R")

#voriconazole physicochemical properties
logP <- 2.56  #lipophilicity
pKa <- 1.76  
fup <- 0.42   #unbound fraction in plasma
type <- 3     #monoprotic base
BP <- 1       #blood:plasma concentration ratio
dat <- read.csv("../data/source/tissue_comp_P&T.csv")

#calculate partition coefficients
Kp <- calcKp_PT(logP=logP, pKa=pKa, fup=fup, BP=BP, type=type, dat=dat)

#update model parameters partition coefficients
modA <- param(modA, Kp)
  1. Simulate the steady state after a 4 mg/kg IV infusion dose of voriconazole given to an adult male with a rate of 4 mg/kg/h twice a day for 7 days. Compare the steady state plasma drug concentration-time profiles from previous simulation to the observed data in Adult_IV.csv. (N.B.: observed data were digitized from Zane and Thakker (2014) paper using WebPlotDigitizer https://automeris.io/WebPlotDigitizer/):
#load observed IV infusion data
obs <- read.csv("../data/source/Adult_IV.csv")

#set simulation conditions
bw   <- 73
amt  <- 4*bw
rate <- 4*bw
cmt  <- "VEN"
ii   <- 12
addl <- 13
ss   <- 1

#run simulation
sim <- 
  modA %>% 
  ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss) %>% 
  mrgsim(delta = 0.1, end = 12) %>% 
  filter(row_number() != 1)  

#plot prediction and compare to observed data
gp <- ggplot() + 
  geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) + 
  geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
  geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) + 
  scale_colour_manual(name='', 
                      values=c('sim'='black', 'observed'='black'), 
                      breaks=c("observed","sim"),
                      labels=c("observed","predicted")) +
  guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
  labs(title="Adult 4 mg/kg IV", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

  1. Simulate the steady state after a 200 mg PO dose of voriconazole given to an adult male with a rate of twice a day for 7 days. Compare the steady state plasma drug concentration-time profile to the observed data in Adult_PO.csv.
obs <- read.csv("../data/source/Adult_PO.csv")

bw   <- 73
amt  <- 200
cmt  <- "GUTLUMEN"
ii   <- 12
addl <- 13
ss   <- 1

sim <- 
  modA %>% 
  ev(amt=amt, cmt=cmt, ii=ii, addl=addl, ss=ss) %>% 
  mrgsim(delta = 0.1, end = 12) %>% 
  filter(row_number() != 1)  

gp <- ggplot() + 
  geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) + 
  geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
  geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) + 
  scale_colour_manual(name='', 
                      values=c('sim'='black', 'observed'='black'), 
                      breaks=c("observed","sim"),
                      labels=c("observed","predicted")) +
  guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
  labs(title="Adult 4 mg/kg PO", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

  1. Integrate the model with an additional gut wall enterocyte compartment to account for intestinal clearance, intestinal transit and lumen solubility effects on absorption rate. Note: use about an intestinal clearance that is 30 times lower than hepatic clearance. Recompile and re-run the previous step. Any change!!

modA <- mread("../model/voriPBPK_ext") %>%
  param(MPPGI = 30.3/30) %>%
  param(Kp)

sim <- 
  modA %>% 
  ev(amt=amt, cmt=cmt, ii=ii, addl=addl, ss=ss) %>% 
  mrgsim(delta = 0.1, end = 12) %>% 
  filter(row_number() != 1)  

gp <- ggplot() + 
  geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) + 
  geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
  geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) + 
  scale_colour_manual(name='', 
                      values=c('sim'='black', 'observed'='black'), 
                      breaks=c("observed","sim"),
                      labels=c("observed","predicted")) +
  guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
  labs(title="Adult 4 mg/kg PO", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

  1. Generate pediatric model
# pediatric (5 yo) male physiology; https://www.ncbi.nlm.nih.gov/pubmed/14506981
pedPhys <- list(WEIGHT = 19,
                Vad = 5.5,
                Vbo = 2.43,
                Vbr = 1.31,
                VguWall = 0.22,
                VguLumen = 0.117,
                Vhe = 0.085,
                Vki = 0.11,
                Vli = 0.467,
                Vlu = 0.125,
                Vmu = 5.6,
                Vsp = 0.05,
                Vbl = 1.5,
                Qad = 0.05*3.4*60,
                Qbo = 0.05*3.4*60,
                Qbr = 0.12*3.4*60,
                Qgu = 0.15*3.4*60, 
                Qhe = 0.04*3.4*60,
                Qki = 0.19*3.4*60,
                Qmu = 0.17*3.4*60,
                Qsp = 0.03*3.4*60,
                Qha = 0.065*3.4*60, 
                Qlu = 3.4*60,
                MPPGL = 26,
                VmaxH = 120.5,
                KmH = 11,
                MPPGI = 0,
                VmaxG = 120.5,
                KmG = 11)

modP <- param(modA, pedPhys)
  1. Simulate the steady state after a 4 mg/kg voriconazole IV infusion dosing in a male child subject infused at a rate of 3 mg/kg/h twice a day for seven days. Compare the steady state to the observed data in Pediatric_IV.csv.
obs <- read.csv("../data/source/Pediatric_IV.csv")  #load observed data

wt   <- 19  #pediatric body weight
amt  <- 4*wt  
rate <- 3*wt
cmt  <- "VEN"  #intravenous infusion
ii   <- 12
addl <- 13
ss   <- 1

# simulate
sim <- 
  modP %>%
  ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=1) %>% 
  mrgsim(delta = 0.1, end = 12) %>% 
  dplyr::filter(row_number() != 1)  

# plot
gp <- ggplot() + 
  geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) + 
  geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
  geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) + 
  scale_colour_manual(name='', 
                      values=c('sim'='black', 'observed'='black'), 
                      breaks=c("observed","sim"),
                      labels=c("observed","predicted")) +
  guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
  labs(title="Pediatric 4 mg/kg IV", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

  1. Simulate the steady state after a 4 mg/kg voriconazole PO dosing in a male child subject twice a day for seven days. Compare to obsreved data in Pediatric_PO.csv Note: Include a similar 30-fold lower intestinal clearance than hepatic clearance.
obs <- read.csv("../data/source/Pediatric_PO.csv")  #load observed data

# adjust intestinal clearance
modP <- modP %>% param(MPPGI = 26 / 30)

# simulation conditions
bw   <- 19
amt  <- 4 * bw
cmt  <- "GUTLUMEN"
ii   <- 12
addl <- 13
ss   <- 1

# simulate
sim <- 
  modP %>%
  ev(amt=amt, cmt=cmt, ii=ii, addl=addl, ss=1) %>% 
  mrgsim(delta = 0.1, end = 12) %>% 
  dplyr::filter(row_number() != 1)  

# plot
gp <- ggplot() + 
  geom_point(data = obs, aes(x=time, y=obs, col="observed"), size=2.5) + 
  geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
  geom_line(data = sim, aes(x=time, y=CP, col="sim"), lwd=1) + 
  scale_colour_manual(name='', 
                      values=c('sim'='black', 'observed'='black'), 
                      breaks=c("observed","sim"),
                      labels=c("observed","predicted")) +
  guides(colour = guide_legend(override.aes = list(linetype=c(0,1), shape=c(16, NA)))) +
  labs(title="Pediatric 4 mg/kg PO", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

3. Sensitivity Analysis

a. Local

Graphical

Run graphical sensitivity analysis for the muscle:plasma (Kpmu) and lung:plasma (Kplu) partition coefficients using adult IV data.

#load observed IV infusion data
obs <- read.csv("../data/source/Adult_IV.csv")

#set simulation conditions
bw   <- 73
amt  <- 4*bw
rate <- 4*bw
cmt  <- "VEN"
ii   <- 12
addl <- 13
ss   <- 1

##' Define an intervention
e <- ev(cmt=cmt, amt=amt, rate=rate, ii= ii, addl=addl, ss=1)

## Sensitivity analysis on Kpmu
idata <- expand.idata(Kpmu = c(3/2, 3, 3*2))

modA %>%
  carry_out(Kpmu) %>% 
  mrgsim_ei(e, idata, delta = 0.1, recsort=3, obsonly=TRUE, end = 12) %>% 
  mutate(Kpmu = factor(Kpmu)) %>%
  ggplot(aes(x=time, y=CP, col=Kpmu)) +
  geom_line() +
  labs(x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()

##' Sensitivity analysis on Kplu
idata <- expand.idata(Kplu = c(1/2, 1, 1*2))

modA %>%
  carry_out(Kplu) %>% 
  mrgsim_ei(e, idata, delta = 0.1, recsort=3, obsonly=TRUE, end = 12) %>% 
  mutate(Kplu = factor(Kplu)) %>%
  ggplot(aes(x=time, y=CP, col=Kplu)) +
  geom_line() +
  labs(x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()

FME package

Use package FME (https://cran.r-project.org/web/packages/FME/index.html) to run local sensitivity analysis for the partition coefficient parameters Soetaert, Karline, and Thomas Petzoldt. 2010. “Inverse Modelling, Sensitivity and Monte Carlo Analysis in R Using Package FME.” Journal of Statistical Software, Articles 33 (3): 1–28. https://cran.r-project.org/web/packages/FME/vignettes/FME.pdf.
What is the most influential parameter?

library(FME)

### sensitivity analysis
#set a list of the candidate parameters
parms <- list(Kpad = 9.89, Kpbo = 7.91, Kpbr = 7.35, Kpgu = 5.82, Kphe = 1.95, Kpki = 2.9, Kpli = 4.66, Kplu = 0.83, Kpmu = 2.94, Kpsp = 2.96) 

#set the output variable of interest
sensvar <- c("CP")  

#set the output function
outFun <- function(pars){
  out <- modA %>%
    param(pars) %>%
    ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss) %>%
    mrgsim(end=12, delta=0.1) %>%
    dplyr::filter(row_number() != 1) 
  
  #get rid of ID because sensFun will take the first column as the time variable
  out <- out %>% dplyr::select(-ID)  
  return(out)
}

locSens <- sensFun(func=outFun, parms=parms, sensvar=sensvar, tiny=1e-4)
summary(locSens)
##      value scale      L1      L2    Mean     Min     Max   N
## Kpad  9.89  9.89 0.00438 0.00510 6.6e-05 -0.0057 0.01191 121
## Kpbo  7.91  7.91 0.00860 0.00994 4.6e-04 -0.0114 0.02109 121
## Kpbr  7.35  7.35 0.00768 0.01049 1.8e-03 -0.0337 0.00753 121
## Kpgu  5.82  5.82 0.00295 0.00415 8.3e-04 -0.0151 0.00375 121
## Kphe  1.95  1.95 0.00072 0.00093 3.0e-04 -0.0041 0.00160 121
## Kpki  2.90  2.90 0.00074 0.00119 1.4e-04 -0.0062 0.00219 121
## Kpli  4.66  4.66 0.00572 0.00837 1.4e-03 -0.0306 0.00699 121
## Kplu  0.83  0.83 0.00029 0.00039 1.1e-04 -0.0013 0.00033 121
## Kpmu  2.94  2.94 0.03825 0.04353 5.3e-03 -0.0818 0.05433 121
## Kpsp  2.96  2.96 0.00037 0.00051 1.2e-04 -0.0020 0.00053 121
## summary plots
plot(summary(locSens))

# nicer view
summ <- as_tibble(summary(locSens)) %>%
  mutate(parms = names(parms)) 

ggplot(data=summ, aes(x=reorder(parms, Mean), y=Mean)) + 
  geom_col() + 
  labs(x="Parameter", y="Coefficient") +
  coord_flip() +
  geom_hline(yintercept = 0, lty=2) +
  theme_bw()

## time-course sensitivity
plot(locSens, which = c("CP"), xlab="time", lwd = 2)

#nicer view
df_temp <- as_tibble(locSens) %>%
  gather(Parameter, Coefficient, -x, -var) %>%
  mutate(Parameter = factor(Parameter)) %>%
  rename(time=x) %>%
  group_by(Parameter) %>%
  mutate(Coefficient = Coefficient - first(Coefficient)) %>%
  ungroup()

ggplot(data=df_temp, aes(x=time, y=Coefficient, col=Parameter)) +
  geom_line() +
  theme(legend.position="right") +
  facet_wrap(~var) +
  theme_bw()

b. Global

Review sobol global sensitivity analysis from sensitivity package (https://cran.r-project.org/web/packages/sensitivity/index.html)
Zhang, X-Y, M. N. Trame, L. J. Lesko, and S. Schmidt. 2015. “Sobol Sensitivity Analysis: A Tool to Guide the Development and Evaluation of Systems Pharmacology Models.” CPT: Pharmacometrics & Systems Pharmacology 4 (2): 69–79. https://ascpt.onlinelibrary.wiley.com/doi/full/10.1002/psp4.6
Vignette: https://github.com/mrgsolve/gallery/blob/master/application/sobol.md

4. Parameter Estimation

library(nloptr)
library(numDeriv)
library(kableExtra)

### optimization with maximum likelihood method
theta <- log(c(Kpmu=2.94, sigma=1))  #initial parameter for MLE; mean and variance
sampl <- obs$time                    #sampling times from observed data

bw <- 73
amt = 4*bw
rate = 4*bw
cmt = "VEN"
ii = 12
addl = 13
ss = 1

##set up objective function
OF <- function(pars, pred=F){
  pars <- lapply(pars,exp)  #Get out of log domain for MLE
  pars <- as.list(pars)
  names(pars) <- names(theta)
  
  ## Get a prediction
  out <- as.data.frame(modA %>% 
                         param(pars) %>%
                         ev(cmt=cmt, amt=amt, rate=rate, ii=ii, addl=addl, ss=ss) %>%
                         mrgsim(end=-1, add=sampl)
  )
  
  out <- out[-1,] 
  
  if(pred) return(out)
  
  ##OLS
  #return(sum((out$Cvenous - obs$obs)^2))
  
  ##maximum likelihood
  return(-1*sum(dnorm(log(obs$obs),
                      mean=log(out$CP),
                      sd=pars$sigma, log=TRUE)))
}

##Fit with nloptr package
##derivative-free optimizers
#fit <- neldermead(theta, OF)  #Nelder-Mead simplex
fit <- newuoa(theta, OF)  #New Unconstrained Optimization with quadratic Approximation

##gradient-basd optimizers
#fit <- tnewton(theta, OF)  #Local optimizer; Nelder-Mead simplex
#fit <- mlsl(theta, OF, lower=log(c(0.1)), upper=log(c(10)))  #global optimizer; multi-level single-linkage; takes very long ~ 10 min

##global optimizers
#fit <- direct(OF, lower=log(c(0.1, 0.1)), upper=log(c(10, 2)))  #DIviding RECTangles algorithm; takes ~ 3 min

fit
## $par
## [1] -0.5709545 -2.0482820
## 
## $value
## [1] -8.181457
## 
## $iter
## [1] 52
## 
## $convergence
## [1] 4
## 
## $message
## [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
p <- as.list(exp(fit$par))  #get the parameters on the linear scale
names(p) <- names(theta)
p
## $Kpmu
## [1] 0.5649859
## 
## $sigma
## [1] 0.1289563
# get standard error and confidence intervals around the estimated parameters
h <- hessian(OF, fit$par)
vc_log <- solve(h)  #variance-covariance matrix
SE_log <- sqrt(diag(vc_log))  #standard error on log scale

# create dataframe with parameter summary
sig <- function(x) signif(x, 3)
paramSumm <- tibble(Parameter = names(theta),
                    Estimate = exp(fit$par),
                    lb = exp(fit$par - (1.96 * SE_log)),
                    ub = exp(fit$par + (1.96 * SE_log))) %>%
  mutate(`Estimate (95% CI)` = paste0(sig(Estimate), " (", sig(lb), ", ", sig(ub), ")")) %>%
  select(Parameter, `Estimate (95% CI)`)

paramSumm %>%
  knitr::kable() %>%
  kable_styling()
Parameter Estimate (95% CI)
Kpmu 0.565 (0.202, 1.58)
sigma 0.129 (0.0878, 0.189) 
# compare initial predictions to those using the optimized parameters
predB4 <- OF(theta, pred=T)
predAfter <- OF(fit$par, pred=T)

gp <- ggplot() +
  geom_point(data=obs, aes(x=time, y=obs)) + 
  geom_errorbar(data=obs, aes(x=time, y=obs, ymin=obs-sd, ymax=obs+sd), width=0) +
  geom_line(data=predB4, aes(x=time, y=CP), lty=2) +
  geom_line(data=predAfter, aes(x=time, y=CP)) +
  labs(x="time (h)", y="Plasma concentration (mg/L)") + 
  theme_bw()
gp

check out:

Simulate a 4 mg/kg voriconazole IV infusion dosing in a male child subject infused at a rate of 3 mg/kg/h twice a day for seven days. Compare the steady state prediction before and after optimization to the observed data in Pediatric_IV.csv.

obs <- read.csv("../data/source/Pediatric_IV.csv")  #load observed data

wt <- 19  #adult body weight
amt <- 4*wt  
rate <- 3*wt
cmt <- "VEN"  #intravenous infusion
ii = 12
addl = 13
ss = 1

# simulate
simB4 <- as.data.frame(modP %>%
                         ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=1) %>% 
                         mrgsim(delta = 0.1, end = 12)) %>% 
  dplyr::filter(row_number() != 1)  

simAfter <- as.data.frame(modP %>%
                            param(Kpmu=p$Kpmu) %>%
                            ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=1) %>% 
                            mrgsim(delta = 0.1, end = 12)) %>% 
  dplyr::filter(row_number() != 1)

# plot
gp <- ggplot() + 
  geom_point(data = obs, aes(x=time, y=obs), size=2.5) + 
  geom_errorbar(data = obs, aes(x = time, y = obs, ymin=obs-sd, ymax=obs+sd), width=0) +
  geom_line(data = simB4, aes(x=time, y=CP), lty=2) + 
  geom_line(data = simAfter, aes(x=time, y=CP)) + 
  labs(title="Pediatric 4 mg/kg IV", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

5. Population simulations

a. How to create a population?

Need to introduce interindividual variability to physiological parameters

Willmann, Stefan, Karsten Höhn, Andrea Edginton, Michael Sevestre, Juri Solodenko, Wolfgang Weiss, Jörg Lippert, and Walter Schmitt. 2007. “Development of a Physiology-Based Whole-Body Population Model for Assessing the Influence of Individual Variability on the Pharmacokinetics of Drugs.” Journal of Pharmacokinetics and Pharmacodynamics 34 (3): 401–31. https://pubmed.ncbi.nlm.nih.gov/17431751/

“NHANES - National Health and Nutrition Examination Survey Homepage.” 2018. July 26, 2018. http://www.cdc.gov/nchs/nhanes/.

Meyer, Michaela, Sebastian Schneckener, Bernd Ludewig, Lars Kuepfer, and Joerg Lippert. 2012. “Using Expression Data for Quantification of Active Processes in Physiologically Based Pharmacokinetic Modeling.” Drug Metabolism and Disposition: The Biological Fate of Chemicals 40 (5): 892–901. https://pubmed.ncbi.nlm.nih.gov/22293118/

b. Simulation

  1. Simulate a simple voriconazole dosing scenario for a population of 100 individuals, 50 males and 50 females, with ages ranging between 20-80, weights ranging between 50-100 kg and heights between 1.5 and 1.9 m. Use population saved in file ../data/derived/popPars_100.rds
# load population params
popPars <- readRDS("../data/derived/popPars_100.rds")

# add IIV on ka and CL/VmaxH
set.seed(192898)
iVmaxH <- rlnorm(100, meanlog=log(40), sdlog=0.2)
ika <- rlnorm(100, meanlog=log(0.849), sdlog=0.2)

# add to popPars
popPars2 <- lapply(1:length(popPars), function(i){
  pars <- c(popPars[[i]],list(VmaxH = iVmaxH[i]))
  pars <- c(pars, list(ka = ika[i]))
  return(pars)
  })

# simulate
modA2 <- mread("../model/voriPBPK2")
Kpmu <- 0.56
modA2 <- param(modA2, Kpmu=Kpmu)

#set simulation conditions
bw   <- 73
amt  <- 4*bw
rate <- 4*bw
cmt  <- "VEN"
ii   <- 12
addl <- 13
ss   <- 1
delta <- 0.1
end <- 12

# prepare simulation dataset
idata <- popPars2 %>% bind_rows()

e <- ev(amt=amt, cmt=cmt, ii=ii, addl=addl, rate=rate, ss=ss)

data <- e %>% 
  as_tibble %>% 
  bind_cols(idata) %>%
  mutate(amt = 4*BW, rate = 4*BW) %>%
  select(ID, everything())

#run simulation
system.time(sims <- modA2 %>% 
  mrgsim_d(data=data, delta = delta, end = end, obsonly=T, outvars = c("CP")) %>% 
  filter(row_number() != 1))
##    user  system elapsed 
##   5.633   0.005   5.642
# get summary stats for population prediction
hi95 <- function(x) quantile(x, probs = c(0.95))
lo05 <- function(x) quantile(x, probs = c(0.05))

sims2 <- sims %>%
  group_by(time) %>%
  mutate(loCP = lo05(CP),
         medCP = median(CP),
         hiCP = hi95(CP)) %>%
  ungroup() %>%
  filter(ID == first(ID))

#plot population predictions 
gp <- ggplot(data = sims2, aes(x=time)) + 
  geom_line(aes(y=medCP), col="black") +
  geom_ribbon(aes(ymin=loCP, ymax=hiCP), alpha = 0.5) +
  scale_y_continuous(trans = "log10") +
  labs(title="Population simulation", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

  1. Use PKNCA package to calculate summary statistics for the population
library(PKNCA)

# get PKNCA object
data_nca <- data %>%
  select(ID, time, amt, rate)

# create dose object
dose_obj <-
  PKNCAdose(
    data,
    amt~time|ID
  )

# create conc object
conc_obj <-
  PKNCAconc(
    sims,
    CP~time|ID
  )

# combine
data_obj <- PKNCAdata(conc_obj, dose_obj)

## Calculate the NCA parameters
results_obj <- pk.nca(data_obj)

## Summarize the results
summary(results_obj)
##  start end   N     auclast        cmax              tmax   half.life
##      0  24 100 26.2 [19.6]           .                 .           .
##      0 Inf 100           . 5.33 [16.7] 1.00 [1.00, 1.00] 27.7 [7.44]
##   aucinf.obs
##            .
##  86.5 [42.6]
## 
## Caption: auclast, cmax, aucinf.obs: geometric mean and geometric coefficient of variation; tmax: median and range; half.life: arithmetic mean and standard deviation
  1. Compare doses 3 and 4 mg/kg for adults. Which dose keeps voriconazole concentration at steady state above MIC (1 mg/L) for >= 90% of subjects?
library(cowplot)

data_3mg <- e %>% 
  as_tibble %>% 
  bind_cols(idata) %>%
  mutate(amt = 3*BW, rate = 3*BW) %>%
  select(ID, everything())

#run simulation
sims_3mg <- modA2 %>% 
  mrgsim_d(data=data_3mg, delta = delta, end = end, obsonly=T, outvars = c("CP")) %>% 
  filter(row_number() != 1)

sims2_3mg <- sims_3mg %>%
  group_by(time) %>%
  mutate(loCP = lo05(CP),
         medCP = median(CP),
         hiCP = hi95(CP)) %>%
  ungroup() %>%
  filter(ID == first(ID))

#plot population predictions 
gp_3mg <- ggplot(data = sims2_3mg, aes(x=time)) + 
  geom_line(aes(y=medCP), col="black") +
  geom_ribbon(aes(ymin=loCP, ymax=hiCP), alpha = 0.5) +
  scale_y_continuous(trans = "log10") +
  labs(title="Population simulation (3 mg/kg)", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw() + 
  geom_hline(aes(yintercept = 1), lty=2) +
  ylim(c(0.1,7))

gp_4mg <- gp + 
  labs(title="Population simulation (4 mg/kg)", x="time (h)", y="Plasma concentration (mg/L)") +
  geom_hline(aes(yintercept = 1), lty=2) +
  ylim(c(0.1,7)) 

gp_34 <- plot_grid(gp_3mg, gp_4mg, ncol=2) 
gp_34

# calculate stats
stats_4mg <- sims %>%
  filter(time != 0) %>%
  group_by(ID) %>%
  mutate(Cmin = min(CP)) %>%
  slice(1) %>%
  ungroup() %>%
  mutate(FLG = ifelse(Cmin < 1, 1, 0))
percLow_4mg <- (nrow(stats_4mg[stats_4mg$FLG==1,]) / nrow(stats_4mg)) * 100

stats_3mg <- sims_3mg %>%
  filter(time != 0) %>%
  group_by(ID) %>%
  mutate(Cmin = min(CP)) %>%
  slice(1) %>%
  ungroup() %>%
  mutate(FLG = ifelse(Cmin < 1, 1, 0))
percLow_3mg <- (nrow(stats_3mg[stats_3mg$FLG==1,]) / nrow(stats_3mg)) * 100 

stats_df <- tibble(Dose = c("3 mg/kg","4 mg/kg"),
                   `Percent of subjects with SS Cmin < MIC` = c(percLow_3mg, percLow_4mg))

stats_df %>%
  knitr::kable() %>%
  kable_styling()
Dose Percent of subjects with SS Cmin < MIC
3 mg/kg 29
4 mg/kg 4

c. Parallelization

Use the package mrgsim.apply (vignette: https://github.com/kylebaron/mrgsim.parallel) to run the same population simulation over parallel cores. How long does it take compared to the single core simulation?

#Sys.setenv(R_LIBS = "../renv/library")
library(parallel)
library(future)
library(mrgsim.parallel)

# set parallelization options
options(future.fork.enable=TRUE, mc.cores = 6L)
plan(multiprocess, workers = 6L)

#run simulation
system.time(sims <- modA2 %>% 
  future_mrgsim_d(data=data, nchunk = 6L, delta = delta, end = end, obsonly=T, outvars = c("CP")) %>% 
  filter(row_number() != 1))
##    user  system elapsed 
##   6.363   0.181   1.401
# get summary stats for population
hi95 <- function(x) quantile(x, probs = c(0.95))
lo05 <- function(x) quantile(x, probs = c(0.05))

sims2 <- sims %>%
  group_by(time) %>%
  mutate(loCP = lo05(CP),
         medCP = median(CP),
         hiCP = hi95(CP)) %>%
  ungroup() %>%
  filter(ID == first(ID))

#plot population predictions 
gp <- ggplot(data = sims2, aes(x=time)) + 
  geom_line(aes(y=medCP), col="black") +
  geom_ribbon(aes(ymin=loCP, ymax=hiCP), alpha = 0.5) +
  scale_y_continuous(trans = "log10") +
  labs(title="Population simulation - parallel", x="time (h)", y="Plasma concentration (mg/L)") +
  theme_bw()
gp

6. Shiny - Interactive Document

Explore the simple shiny app saved as app.R

..and this is why modeling PBPK in R beats anything else..